Question

If $0<A<\pi /2$ the value of the expression $\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}-secAcosecA$ is equal to

• A. -1
• B. 0
• C. 1
• D. sin A + cos A

Answer 1

The correct option is C 1

$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}-\sec AcosecA$
$=\frac{{\sin }^{2}A}{\cos A(\sin A-\cos A)}+\frac{{\cos }^{2}A}{\sin A(\cos A-\sin A)}-\sec AcosecA$
$=\frac{{\sin }^{3}A-{\cos }^{3}A}{\sin A\cos A(\sin A-\cos A)}-\sec AcosecA$
$=\frac{{\sin }^{2}A+{\cos }^{2}A+\sin A\cos A}{\sin A\cos A}-\sec AcosecA$ $[a^3-b^3=(a-b\left)\right(a^2+ab+b^2\left)\right]$
$=\sec AcosecA+1-\sec AcosecA$

$=1$.